Interactive Purchasing Situations

Abstract: This paper introduces a new class of interactive cooperative purchasing situations and provides an explicit alternative characterization of the nucleolus of cooperative games, which offers an alternative to Kohlberg (1971). In our cooperative purchasing situation, the unit price of a commodity depends on the largest order quantity within a cooperating group of players. Due to a decreasing unit price function, players can obtain cost savings by purchasing cooperatively. However, to establish fruitful cooperation a decision has to be made about an adequate allocation of the corresponding cost savings. We analyze Maximum Cooperative Purchasing (MCP)-situation from the perspective of allocation by defining corresponding cooperative MCP-games. It turns out that in an MCP-game all coalitional values can be determined from the values of two-player coalitions. Moreover, it is shown that a decreasing unit price function is a sufficient condition for a non-empty core: the Direct Price solution is both a core element and a marginal vector. It is seen that the nucleolus of an MCP-game can be derived in polynomial time from the Direct Price solution, using a socalled nucleolus determinant. To show this result, the explicit alternative characterization of the nucleolus is used. Using the decomposition of an MCP-game into unanimity games we find an explicit expression for the Shapley value. Interestingly, the Shapley value can be interpreted as a specific tax and subsidize system. Finally, the behavior of the three solution concepts is compared numerically.